The previous admin installed Backupexec on our file server/domain controller, which makes me worry that if this server fails I won't be able to do a restore because the local BackupExec database will be lost. Is this correct? Or will I be able to just install it on a seperate computer and point it towards our NAS and it will magically see all the files? I'm assuming BE should be installed on its own server. Thanks.
Assuming you're covering the Backup Exec folder with your backup you can "bare metal" restore Backup Exec from only backup media. When you say "...point it towards our NAS..." it makes me think you're saving backups in a Backup-to-Disk folder on a NAS device. In a disaster scenario (assuming your building didn't burn taking the NAS device with it) where the Backup Exec server failed you would be able to install Backup Exec on another server, add the Backup-to-Disk folder on the NAS to the Backup Exec server, and catalog the media on the NAS to obtain a list of backup sets to restore.
At some level your question is "if the server running Backup Exec fails, can I recover my backups?" Whether the server in question is a DC or file server is irrelevant.
And the answer is "yes, you can". Basically you stand up a new server and install Backup Exec on it (or install Backup Exec on another existing server), inventory the media you're backing up to, and Backup Exec will pick up what files/folders/whatever you've backed up. Then you can restore it to its appropriate place.
Ideally every application/service would have its own server. Realistically that never happens. Throw Backup Exec wherever you want/can shoehorn it in properly. It'll work fine.